LMFDB - Embedded newform 1503.1.f.b.1168.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,1,Mod(166,1503)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1503.166");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1503.f (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.750094713987$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} + \cdots)$

Embedding invariants

Embedding label			1168.3
Root			$0.928368 - 0.371662i$ of defining polynomial
Character	$\chi$	$=$	1503.1168
Dual form			1503.1.f.b.166.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.723734 + 1.25354i) q^{2} +(0.580057 + 0.814576i) q^{3} +(-0.547582 - 0.948440i) q^{4} +(-1.44091 + 0.137591i) q^{6} +(0.786053 - 1.36148i) q^{7} +0.137747 q^{8} +(-0.327068 + 0.945001i) q^{9} +O(q^{10})$	\(q+(-0.723734 + 1.25354i) q^{2} +(0.580057 + 0.814576i) q^{3} +(-0.547582 - 0.948440i) q^{4} +(-1.44091 + 0.137591i) q^{6} +(0.786053 - 1.36148i) q^{7} +0.137747 q^{8} +(-0.327068 + 0.945001i) q^{9} +(0.888835 - 1.53951i) q^{11} +(0.454947 - 0.996196i) q^{12} +(1.13779 + 1.97070i) q^{14} +(0.447890 - 0.775768i) q^{16} +(-0.947890 - 1.09392i) q^{18} +1.96386 q^{19} +(1.56499 - 0.149438i) q^{21} +(1.28656 + 2.22839i) q^{22} +(0.0799009 + 0.112205i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.959493 + 0.281733i) q^{27} -1.72171 q^{28} +(-0.928368 + 1.60798i) q^{29} +(-0.723734 - 1.25354i) q^{31} +(0.717180 + 1.24219i) q^{32} +(1.76962 - 0.168978i) q^{33} +(1.07537 - 0.207261i) q^{36} +(-1.42131 + 2.46178i) q^{38} +(-0.945307 + 2.06993i) q^{42} -1.94684 q^{44} +(0.327068 - 0.566498i) q^{47} +(0.891724 - 0.0851493i) q^{48} +(-0.735759 - 1.27437i) q^{49} +(-0.723734 - 1.25354i) q^{50} +(0.341254 - 1.40667i) q^{54} +(0.108276 - 0.187540i) q^{56} +(1.13915 + 1.59971i) q^{57} +(-1.34378 - 2.32750i) q^{58} +(-0.415415 + 0.719520i) q^{61} +2.09516 q^{62} +(1.02951 + 1.18812i) q^{63} -1.18041 q^{64} +(-1.06891 + 2.34059i) q^{66} +(-0.0450525 + 0.130171i) q^{72} +(-0.995472 + 0.0950560i) q^{75} +(-1.07537 - 1.86260i) q^{76} +(-1.39734 - 2.42027i) q^{77} +(-0.786053 - 0.618159i) q^{81} +(-0.998692 - 1.40247i) q^{84} +(-1.84833 + 0.176494i) q^{87} +(0.122434 - 0.212062i) q^{88} +1.16011 q^{89} +(0.601300 - 1.31666i) q^{93} +(0.473420 + 0.819988i) q^{94} +(-0.595855 + 1.30474i) q^{96} +(-0.841254 + 1.45709i) q^{97} +2.12998 q^{98} +(1.16413 + 1.34347i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) $ 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9	$ 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9} - q^{11} + q^{14} - 12 q^{16} + 2 q^{18} + 2 q^{19} - q^{21} + q^{22} + q^{24} - 10 q^{25} - 2 q^{27} - q^{29} - q^{31} - q^{33} + q^{38} - 13 q^{42} - 22 q^{44} - q^{47} + 21 q^{48} - 11 q^{49} - q^{50} - 12 q^{54} - q^{56} - q^{57} + q^{58} + 2 q^{61} + 42 q^{62} + 2 q^{63} + 20 q^{64} - 2 q^{66} - 10 q^{72} + q^{75} + q^{77} + q^{81} + 22 q^{84} - q^{87} - q^{88} + 2 q^{89} + 2 q^{93} + q^{94} + 2 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) $ 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9 - q^11 + q^14 - 12 * q^16 + 2 * q^18 + 2 * q^19 - q^21 + q^22 + q^24 - 10 * q^25 - 2 * q^27 - q^29 - q^31 - q^33 + q^38 - 13 * q^42 - 22 * q^44 - q^47 + 21 * q^48 - 11 * q^49 - q^50 - 12 * q^54 - q^56 - q^57 + q^58 + 2 * q^61 + 42 * q^62 + 2 * q^63 + 20 * q^64 - 2 * q^66 - 10 * q^72 + q^75 + q^77 + q^81 + 22 * q^84 - q^87 - q^88 + 2 * q^89 + 2 * q^93 + q^94 + 2 * q^97 - 22 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	0.235759	+	0.971812i	$0.424242\pi$
$2$	−0.723734	+	1.25354i	−0.723734	+	1.25354i	−0.959493	+	0.281733i	$0.909091\pi$
$3$	0.580057	+	0.814576i	0.580057	+	0.814576i
$3$	0.580057	+	0.814576i	0.580057	+	0.814576i
$4$	−0.547582	−	0.948440i	−0.547582	−	0.948440i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	−1.44091	+	0.137591i	−1.44091	+	0.137591i
$7$	0.786053	−	1.36148i	0.786053	−	1.36148i	−0.142315	−	0.989821i	$-0.545455\pi$
$7$	0.786053	−	1.36148i	0.786053	−	1.36148i	0.928368	−	0.371662i	$-0.121212\pi$
$8$	0.137747			0.137747
$9$	−0.327068	+	0.945001i	−0.327068	+	0.945001i
$10$		0			0
$11$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.0475819	−	0.998867i	$-0.484848\pi$
$11$	0.888835	−	1.53951i	0.888835	−	1.53951i	0.841254	−	0.540641i	$-0.181818\pi$
$12$	0.454947	−	0.996196i	0.454947	−	0.996196i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$	1.13779	+	1.97070i	1.13779	+	1.97070i
$15$		0			0
$16$	0.447890	−	0.775768i	0.447890	−	0.775768i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.947890	−	1.09392i	−0.947890	−	1.09392i
$19$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$19$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$20$		0			0
$21$	1.56499	−	0.149438i	1.56499	−	0.149438i
$22$	1.28656	+	2.22839i	1.28656	+	2.22839i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	0.0799009	+	0.112205i	0.0799009	+	0.112205i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$28$	−1.72171			−1.72171
$29$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$29$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$30$		0			0
$31$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	−0.959493	−	0.281733i	$-0.909091\pi$
$31$	−0.723734	−	1.25354i	−0.723734	−	1.25354i	0.235759	−	0.971812i	$-0.424242\pi$
$32$	0.717180	+	1.24219i	0.717180	+	1.24219i
$33$	1.76962	−	0.168978i	1.76962	−	0.168978i
$34$		0			0
$35$		0			0
$36$	1.07537	−	0.207261i	1.07537	−	0.207261i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−1.42131	+	2.46178i	−1.42131	+	2.46178i
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$	−0.945307	+	2.06993i	−0.945307	+	2.06993i
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	−1.94684			−1.94684
$45$		0			0
$46$		0			0
$47$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$47$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$48$	0.891724	−	0.0851493i	0.891724	−	0.0851493i
$49$	−0.735759	−	1.27437i	−0.735759	−	1.27437i
$50$	−0.723734	−	1.25354i	−0.723734	−	1.25354i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.341254	−	1.40667i	0.341254	−	1.40667i
$55$		0			0
$56$	0.108276	−	0.187540i	0.108276	−	0.187540i
$57$	1.13915	+	1.59971i	1.13915	+	1.59971i
$58$	−1.34378	−	2.32750i	−1.34378	−	2.32750i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$61$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$62$	2.09516			2.09516
$63$	1.02951	+	1.18812i	1.02951	+	1.18812i
$64$	−1.18041			−1.18041
$65$		0			0
$66$	−1.06891	+	2.34059i	−1.06891	+	2.34059i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.0450525	+	0.130171i	−0.0450525	+	0.130171i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$76$	−1.07537	−	1.86260i	−1.07537	−	1.86260i
$77$	−1.39734	−	2.42027i	−1.39734	−	2.42027i
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−0.786053	−	0.618159i	−0.786053	−	0.618159i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$	−0.998692	−	1.40247i	−0.998692	−	1.40247i
$85$		0			0
$86$		0			0
$87$	−1.84833	+	0.176494i	−1.84833	+	0.176494i
$88$	0.122434	−	0.212062i	0.122434	−	0.212062i
$89$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$89$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.601300	−	1.31666i	0.601300	−	1.31666i
$94$	0.473420	+	0.819988i	0.473420	+	0.819988i
$95$		0			0
$96$	−0.595855	+	1.30474i	−0.595855	+	1.30474i
$97$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$97$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$98$	2.12998			2.12998
$99$	1.16413	+	1.34347i	1.16413	+	1.34347i
$100$	1.09516			1.09516
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$107$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$108$	0.792607	+	0.755750i	0.792607	+	0.755750i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−0.704131	−	1.21959i	−0.704131	−	1.21959i
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$	−2.82975	+	0.270208i	−2.82975	+	0.270208i
$115$		0			0
$116$	2.03343			2.03343
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.08006	−	1.87071i	−1.08006	−	1.87071i
$122$	−0.601300	−	1.04148i	−0.601300	−	1.04148i
$123$		0			0
$124$	−0.792607	+	1.37284i	−0.792607	+	1.37284i
$125$		0			0
$126$	−2.23445	+	0.430655i	−2.23445	+	0.430655i
$127$	−1.77767			−1.77767			−0.888835	−	0.458227i	$-0.848485\pi$
$127$	−1.77767			−1.77767			−0.888835	+	0.458227i	$0.848485\pi$
$128$	0.137123	−	0.237504i	0.137123	−	0.237504i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−1.12928	−	1.58585i	−1.12928	−	1.58585i
$133$	1.54370	−	2.67376i	1.54370	−	2.67376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$137$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$	0.651174	−	0.0621796i	0.651174	−	0.0621796i
$142$		0			0
$143$		0			0
$144$	0.586611	+	0.676985i	0.586611	+	0.676985i
$145$		0			0
$146$		0			0
$147$	0.611291	−	1.33854i	0.611291	−	1.33854i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	0.601300	−	1.31666i	0.601300	−	1.31666i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	0.270515			0.270515
$153$		0			0
$154$	4.04522			4.04522
$155$		0			0
$156$		0			0
$157$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$157$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1.34378	−	0.537970i	1.34378	−	0.537970i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$168$	0.215572	−	0.0205846i	0.215572	−	0.0205846i
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$	−0.642315	+	1.85585i	−0.642315	+	1.85585i
$172$		0			0
$173$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	0.0475819	+	0.998867i	$0.484848\pi$
$173$	−0.841254	+	1.45709i	−0.841254	+	1.45709i	−0.888835	+	0.458227i	$0.848485\pi$
$174$	1.11646	−	2.44470i	1.11646	−	2.44470i
$175$	0.786053	+	1.36148i	0.786053	+	1.36148i
$176$	−0.796201	−	1.37906i	−0.796201	−	1.37906i
$177$		0			0
$178$	−0.839614	+	1.45425i	−0.839614	+	1.45425i
$179$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$179$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$180$		0			0
$181$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$181$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$182$		0			0
$183$	−0.827068	+	0.0789754i	−0.827068	+	0.0789754i
$184$		0			0
$185$		0			0
$186$	1.21531	+	1.70667i	1.21531	+	1.70667i
$187$		0			0
$188$	−0.716386			−0.716386
$189$	−0.370638	+	1.52779i	−0.370638	+	1.52779i
$190$		0			0
$191$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$191$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$192$	−0.684705	−	0.961533i	−0.684705	−	0.961533i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	−1.21769	−	2.10910i	−1.21769	−	2.10910i
$195$		0			0
$196$	−0.805777	+	1.39565i	−0.805777	+	1.39565i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−2.52662	+	0.486967i	−2.52662	+	0.486967i
$199$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$199$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$200$	−0.0688733	+	0.119292i	−0.0688733	+	0.119292i
$201$		0			0
$202$		0			0
$203$	1.45949	+	2.52792i	1.45949	+	2.52792i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.74555	−	3.02337i	1.74555	−	3.02337i
$210$		0			0
$211$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$211$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$212$		0			0
$213$		0			0
$214$	1.13779	−	1.97070i	1.13779	−	1.97070i
$215$		0			0
$216$	−0.132167	+	0.0388077i	−0.132167	+	0.0388077i
$217$	−2.27557			−2.27557
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$223$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$224$	2.25497			2.25497
$225$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	0.893452	−	1.95639i	0.893452	−	1.95639i
$229$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$229$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$230$		0			0
$231$	1.16096	−	2.54214i	1.16096	−	2.54214i
$232$	−0.127880	+	0.221494i	−0.127880	+	0.221494i
$233$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$233$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	−0.995472	−	0.0950560i	$-0.969697\pi$
$239$	−0.580057	−	1.00469i	−0.580057	−	1.00469i	0.415415	−	0.909632i	$-0.363636\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	3.12670			3.12670
$243$	0.0475819	−	0.998867i	0.0475819	−	0.998867i
$244$	0.909895			0.909895
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−0.0996919	−	0.172671i	−0.0996919	−	0.172671i
$249$		0			0
$250$		0			0
$251$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$251$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$252$	0.563117	−	1.62702i	0.563117	−	1.62702i
$253$		0			0
$254$	1.28656	−	2.22839i	1.28656	−	2.22839i
$255$		0			0
$256$	−0.391724	−	0.678486i	−0.391724	−	0.678486i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1.21590	−	1.40323i	−1.21590	−	1.40323i
$262$		0			0
$263$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$263$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$264$	0.243759	−	0.0232762i	0.243759	−	0.0232762i
$265$		0			0
$266$	2.23445	+	3.87018i	2.23445	+	3.87018i
$267$	0.672932	+	0.945001i	0.672932	+	0.945001i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	−0.0688733	−	0.119292i	−0.0688733	−	0.119292i
$275$	0.888835	+	1.53951i	0.888835	+	1.53951i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$	1.42131	−	0.273935i	1.42131	−	0.273935i
$280$		0			0
$281$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$281$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$282$	−0.393332	+	0.861277i	−0.393332	+	0.861277i
$283$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	−0.888835	−	0.458227i	$-0.848485\pi$
$283$	−0.841254	−	1.45709i	−0.841254	−	1.45709i	0.0475819	−	0.998867i	$-0.484848\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.40844	+	0.271454i	−1.40844	+	0.271454i
$289$	1.00000			1.00000
$290$		0			0
$291$	−1.67489	+	0.159932i	−1.67489	+	0.159932i
$292$		0			0
$293$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$293$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$294$	1.23551	+	1.73503i	1.23551	+	1.73503i
$295$		0			0
$296$		0			0
$297$	−0.419102	+	1.72756i	−0.419102	+	1.72756i
$298$		0			0
$299$		0			0
$300$	0.635257	+	0.892094i	0.635257	+	0.892094i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.879592	−	1.52350i	0.879592	−	1.52350i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−1.53032	+	2.65059i	−1.53032	+	2.65059i
$309$		0			0
$310$		0			0
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	−2.77767			−2.77767
$315$		0			0
$316$		0			0
$317$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$317$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$318$		0			0
$319$	1.65033	+	2.85846i	1.65033	+	2.85846i
$320$		0			0
$321$	−0.911911	−	1.28060i	−0.911911	−	1.28060i
$322$		0			0
$323$		0			0
$324$	−0.155858	+	1.08402i	−0.155858	+	1.08402i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.514186	−	0.890596i	−0.514186	−	0.890596i
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$	1.44747			1.44747
$335$		0			0
$336$	0.585013	−	1.28100i	0.585013	−	1.28100i
$337$	0.327068	+	0.566498i	0.327068	+	0.566498i	0.981929	−	0.189251i	$-0.0606061\pi$
$337$	0.327068	+	0.566498i	0.327068	+	0.566498i	−0.654861	+	0.755750i	$0.727273\pi$
$338$	−0.723734	−	1.25354i	−0.723734	−	1.25354i
$339$		0			0
$340$		0			0
$341$	−2.57312			−2.57312
$342$	−1.86152	−	2.14831i	−1.86152	−	2.14831i
$343$	−0.741276			−0.741276
$344$		0			0
$345$		0			0
$346$	−1.21769	−	2.10910i	−1.21769	−	2.10910i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	1.17951	+	1.65638i	1.17951	+	1.65638i
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$	−2.27557			−2.27557
$351$		0			0
$352$	2.54982			2.54982
$353$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$353$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$354$		0			0
$355$		0			0
$356$	−0.635257	−	1.10030i	−0.635257	−	1.10030i
$357$		0			0
$358$	−1.42131	+	2.46178i	−1.42131	+	2.46178i
$359$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$359$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$360$		0			0
$361$	2.85674			2.85674
$362$	0.473420	−	0.819988i	0.473420	−	0.819988i
$363$	0.897344	−	1.96491i	0.897344	−	1.96491i
$364$		0			0
$365$		0			0
$366$	0.499578	−	1.09392i	0.499578	−	1.09392i
$367$	0.327068	−	0.566498i	0.327068	−	0.566498i	−0.654861	−	0.755750i	$-0.727273\pi$
$367$	0.327068	−	0.566498i	0.327068	−	0.566498i	0.981929	+	0.189251i	$0.0606061\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.57804	+	0.150684i	−1.57804	+	0.150684i
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	0.0450525	−	0.0780332i	0.0450525	−	0.0780332i
$377$		0			0
$378$	−1.64691	−	1.57033i	−1.64691	−	1.57033i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.03115	−	1.44805i	−1.03115	−	1.44805i
$382$	−0.341254	−	0.591068i	−0.341254	−	0.591068i
$383$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$383$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$384$	0.273004	−	0.0260687i	0.273004	−	0.0260687i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	1.84262			1.84262
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	−0.101348	−	0.175540i	−0.101348	−	0.175540i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.636749	−	1.83977i	0.636749	−	1.83977i
$397$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$397$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$398$	−1.21769	+	2.10910i	−1.21769	+	2.10910i
$399$	3.07341	−	0.293475i	3.07341	−	0.293475i
$400$	0.447890	+	0.775768i	0.447890	+	0.775768i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−4.22514			−4.22514
$407$		0			0
$408$		0			0
$409$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.841254	+	0.540641i	$0.181818\pi$
$409$	0.888835	+	1.53951i	0.888835	+	1.53951i	0.0475819	+	0.998867i	$0.484848\pi$
$410$		0			0
$411$	−0.0947329	+	0.00904590i	−0.0947329	+	0.00904590i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	2.52662	+	4.37624i	2.52662	+	4.37624i
$419$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$419$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$420$		0			0
$421$	0.142315	−	0.246497i	0.142315	−	0.246497i	−0.786053	−	0.618159i	$-0.787879\pi$
$421$	0.142315	−	0.246497i	0.142315	−	0.246497i	0.928368	+	0.371662i	$0.121212\pi$
$422$	−2.77767			−2.77767
$423$	0.428368	+	0.494363i	0.428368	+	0.494363i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.653077	+	1.13116i	0.653077	+	1.13116i
$428$	0.860857	+	1.49105i	0.860857	+	1.49105i
$429$		0			0
$430$		0			0
$431$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$431$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$432$	−0.211188	+	0.870529i	−0.211188	+	0.870529i
$433$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$433$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$434$	1.64691	−	2.85253i	1.64691	−	2.85253i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	1.44493	−	0.278487i	1.44493	−	0.278487i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$	−0.839614	−	1.45425i	−0.839614	−	1.45425i
$447$		0			0
$448$	−0.927865	+	1.60711i	−0.927865	+	1.60711i
$449$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$449$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$450$	1.42131	−	0.273935i	1.42131	−	0.273935i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	0.156914	+	0.220355i	0.156914	+	0.220355i
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	2.68757			2.68757
$459$		0			0
$460$		0			0
$461$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$461$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$462$	2.34646	+	3.29514i	2.34646	+	3.29514i
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	0.831613	+	1.44040i	0.831613	+	1.44040i
$465$		0			0
$466$	−0.341254	+	0.591068i	−0.341254	+	0.591068i
$467$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$467$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.981929	+	1.70075i	−0.981929	+	1.70075i
$476$		0			0
$477$		0			0
$478$	1.67923			1.67923
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.18284	+	2.04874i	−1.18284	+	2.04874i
$485$		0			0
$486$	1.21769	+	0.782560i	1.21769	+	0.782560i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−0.0572220	+	0.0991114i	−0.0572220	+	0.0991114i
$489$		0			0
$490$		0			0
$491$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.29661			−1.29661
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$	0.415415	−	0.909632i	0.415415	−	0.909632i
$502$	0.205996	−	0.356796i	0.205996	−	0.356796i
$503$	−1.99094			−1.99094			−0.995472	−	0.0950560i	$-0.969697\pi$
$503$	−1.99094			−1.99094			−0.995472	+	0.0950560i	$0.969697\pi$
$504$	0.141812	+	0.163659i	0.141812	+	0.163659i
$505$		0			0
$506$		0			0
$507$	−0.995472	+	0.0950560i	−0.995472	+	0.0950560i
$508$	0.973420	+	1.68601i	0.973420	+	1.68601i
$509$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	0.841254	−	0.540641i	$-0.181818\pi$
$509$	−0.0475819	−	0.0824143i	−0.0475819	−	0.0824143i	−0.888835	+	0.458227i	$0.848485\pi$
$510$		0			0
$511$		0			0
$512$	1.40826			1.40826
$513$	−1.88431	+	0.553283i	−1.88431	+	0.553283i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.581419	−	1.00705i	−0.581419	−	1.00705i
$518$		0			0
$519$	−1.67489	+	0.159932i	−1.67489	+	0.159932i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	2.63900	−	0.508625i	2.63900	−	0.508625i
$523$	0.471518			0.471518			0.235759	−	0.971812i	$-0.424242\pi$
$523$	0.471518			0.471518			0.235759	+	0.971812i	$0.424242\pi$
$524$		0			0
$525$	−0.653077	+	1.43004i	−0.653077	+	1.43004i
$526$	−0.601300	−	1.04148i	−0.601300	−	1.04148i
$527$		0			0
$528$	0.661508	−	1.44850i	0.661508	−	1.44850i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−3.38120			−3.38120
$533$		0			0
$534$	−1.67162	+	0.159621i	−1.67162	+	0.159621i
$535$		0			0
$536$		0			0
$537$	1.13915	+	1.59971i	1.13915	+	1.59971i
$538$		0			0
$539$	−2.61587			−2.61587
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−0.379436	−	0.532843i	−0.379436	−	0.532843i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	0.104220			0.104220
$549$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$550$	−2.57312			−2.57312
$551$	−1.82318	+	3.15784i	−1.82318	+	3.15784i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.0951638			0.0951638			0.0475819	−	0.998867i	$-0.484848\pi$
$557$	0.0951638			0.0951638			0.0475819	+	0.998867i	$0.484848\pi$
$558$	−0.685261	+	1.97993i	−0.685261	+	1.97993i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.44091	+	2.49574i	1.44091	+	2.49574i
$563$	0.654861	+	1.13425i	0.654861	+	1.13425i	0.981929	+	0.189251i	$0.0606061\pi$
$563$	0.654861	+	1.13425i	0.654861	+	1.13425i	−0.327068	+	0.945001i	$0.606061\pi$
$564$	−0.415545	−	0.583551i	−0.415545	−	0.583551i
$565$		0			0
$566$	2.43538			2.43538
$567$	−1.45949	+	0.584293i	−1.45949	+	0.584293i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$	−0.469383	+	0.0448206i	−0.469383	+	0.0448206i
$574$		0			0
$575$		0			0
$576$	0.386074	−	1.11549i	0.386074	−	1.11549i
$577$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$577$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$578$	−0.723734	+	1.25354i	−0.723734	+	1.25354i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	1.01169	−	2.21530i	1.01169	−	2.21530i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.682507			0.682507
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	−1.60426	+	0.153188i	−1.60426	+	0.153188i
$589$	−1.42131	−	2.46178i	−1.42131	−	2.46178i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−1.86226	−	1.77566i	−1.86226	−	1.77566i
$595$		0			0
$596$		0			0
$597$	0.975950	+	1.37053i	0.975950	+	1.37053i
$598$		0			0
$599$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$599$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$600$	−0.137123	+	0.0130936i	−0.137123	+	0.0130936i
$601$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$601$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	1.40844	+	2.43949i	1.40844	+	2.43949i
$609$	−1.21259	+	2.65520i	−1.21259	+	2.65520i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$613$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$614$		0			0
$615$		0			0
$616$	−0.192479	−	0.333384i	−0.192479	−	0.333384i
$617$	0.142315	+	0.246497i	0.142315	+	0.246497i	0.928368	−	0.371662i	$-0.121212\pi$
$617$	0.142315	+	0.246497i	0.142315	+	0.246497i	−0.786053	+	0.618159i	$0.787879\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$	−1.44747			−1.44747
$623$	0.911911	−	1.57948i	0.911911	−	1.57948i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	3.47528	−	0.331849i	3.47528	−	0.331849i
$628$	1.05080	−	1.82004i	1.05080	−	1.82004i
$629$		0			0
$630$		0			0
$631$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$631$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$632$		0			0
$633$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$634$	1.44091	+	2.49574i	1.44091	+	2.49574i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−4.77761			−4.77761
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	2.26527	−	0.216307i	2.26527	−	0.216307i
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.108276	−	0.0851493i	−0.108276	−	0.0851493i
$649$		0			0
$650$		0			0
$651$	−1.31996	−	1.85363i	−1.31996	−	1.85363i
$652$		0			0
$653$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.723734	+	0.690079i	$0.242424\pi$
$653$	0.959493	+	1.66189i	0.959493	+	1.66189i	0.235759	+	0.971812i	$0.424242\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	1.48853			1.48853
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.547582	+	0.948440i	−0.547582	+	0.948440i
$669$	−1.15486	+	0.110276i	−1.15486	+	0.110276i
$670$		0			0
$671$	0.738471	+	1.27907i	0.738471	+	1.27907i
$672$	1.30801	+	1.83684i	1.30801	+	1.83684i
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$	−0.946841			−0.946841
$675$	0.235759	−	0.971812i	0.235759	−	0.971812i
$676$	1.09516			1.09516
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$678$		0			0
$679$	1.32254	+	2.29071i	1.32254	+	2.29071i
$680$		0			0
$681$		0			0
$682$	1.86226	−	3.22552i	1.86226	−	3.22552i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	2.11188	−	0.407031i	2.11188	−	0.407031i
$685$		0			0
$686$	0.536487	−	0.929222i	0.536487	−	0.929222i
$687$	0.771316	−	1.68895i	0.771316	−	1.68895i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	1.84262			1.84262
$693$	2.74418	−	0.528898i	2.74418	−	0.528898i
$694$		0			0
$695$		0			0
$696$	−0.254601	+	0.0243114i	−0.254601	+	0.0243114i
$697$		0			0
$698$		0			0
$699$	0.273507	+	0.384087i	0.273507	+	0.384087i
$700$	0.860857	−	1.49105i	0.860857	−	1.49105i
$701$	1.16011			1.16011			0.580057	−	0.814576i	$-0.303030\pi$
$701$	1.16011			1.16011			0.580057	+	0.814576i	$0.303030\pi$
$702$		0			0
$703$		0			0
$704$	−1.04919	+	1.81725i	−1.04919	+	1.81725i
$705$		0			0
$706$	0.205996	+	0.356796i	0.205996	+	0.356796i
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	0.159802			0.159802
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.07537	−	1.86260i	−1.07537	−	1.86260i
$717$	0.481929	−	1.05528i	0.481929	−	1.05528i
$718$	1.38884	−	2.40553i	1.38884	−	2.40553i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−2.06752	+	3.58104i	−2.06752	+	3.58104i
$723$		0			0
$724$	0.358193	+	0.620408i	0.358193	+	0.620408i
$725$	−0.928368	−	1.60798i	−0.928368	−	1.60798i
$726$	1.81366	+	2.54693i	1.81366	+	2.54693i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	0.841254	−	0.540641i	0.841254	−	0.540641i
$730$		0			0
$731$		0			0
$732$	0.527791	+	0.741179i	0.527791	+	0.741179i
$733$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$733$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$734$	0.473420	+	0.819988i	0.473420	+	0.819988i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$744$	0.0828270	−	0.181366i	0.0828270	−	0.181366i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.23576	+	2.14040i	−1.23576	+	2.14040i
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	−0.292981	−	0.507458i	−0.292981	−	0.507458i
$753$	−0.165101	−	0.231852i	−0.165101	−	0.231852i
$754$		0			0
$755$		0			0
$756$	1.65197	−	0.485063i	1.65197	−	0.485063i
$757$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$757$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	0.580057	−	0.814576i	$-0.303030\pi$
$761$	−0.415415	−	0.719520i	−0.415415	−	0.719520i	−0.995472	+	0.0950560i	$0.969697\pi$
$762$	2.56147	−	0.244591i	2.56147	−	0.244591i
$763$		0			0
$764$	0.516389			0.516389
$765$		0			0
$766$	−0.411992			−0.411992
$767$		0			0
$768$	0.325456	−	0.712649i	0.325456	−	0.712649i
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	1.44747			1.44747
$776$	−0.115880	+	0.200710i	−0.115880	+	0.200710i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.437742	−	1.80440i	0.437742	−	1.80440i
$784$	−1.31816			−1.31816
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$	−0.827068	+	0.0789754i	−0.827068	+	0.0789754i
$790$		0			0
$791$		0			0
$792$	0.160355	+	0.185059i	0.160355	+	0.185059i
$793$		0			0
$794$	1.44091	−	2.49574i	1.44091	−	2.49574i
$795$		0			0
$796$	−0.921310	−	1.59576i	−0.921310	−	1.59576i
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$	−1.85645	+	4.06506i	−1.85645	+	4.06506i
$799$		0			0
$800$	−1.43436			−1.43436
$801$	−0.379436	+	1.09631i	−0.379436	+	1.09631i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$809$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	1.59838	−	2.76848i	1.59838	−	2.76848i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−2.57312			−2.57312
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	0.0572220	−	0.125299i	0.0572220	−	0.125299i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	−0.738471	+	1.61703i	−0.738471	+	1.61703i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−3.82332			−3.82332
$837$	1.04758	+	0.998867i	1.04758	+	0.998867i
$838$	0.682507			0.682507
$839$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	−0.995472	−	0.0950560i	$-0.969697\pi$
$839$	−0.415415	+	0.719520i	−0.415415	+	0.719520i	0.580057	+	0.814576i	$0.303030\pi$
$840$		0			0
$841$	−1.22373	−	2.11957i	−1.22373	−	2.11957i
$842$	0.205996	+	0.356796i	0.205996	+	0.356796i
$843$	1.98193	−	0.189251i	1.98193	−	0.189251i
$844$	1.05080	−	1.82004i	1.05080	−	1.82004i
$845$		0			0
$846$	−0.929730	+	0.179191i	−0.929730	+	0.179191i
$847$	−3.39593			−3.39593
$848$		0			0
$849$	0.698939	−	1.53046i	0.698939	−	1.53046i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	−0.888835	−	0.458227i	$-0.848485\pi$
$853$	−0.0475819	+	0.0824143i	−0.0475819	+	0.0824143i	0.841254	+	0.540641i	$0.181818\pi$
$854$	−1.89061			−1.89061
$855$		0			0
$856$	−0.216552			−0.216552
$857$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.327068	+	0.945001i	$0.606061\pi$
$857$	−0.981929	+	1.70075i	−0.981929	+	1.70075i	−0.654861	+	0.755750i	$0.727273\pi$
$858$		0			0
$859$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.786053	−	0.618159i	$-0.787879\pi$
$859$	−0.928368	−	1.60798i	−0.928368	−	1.60798i	−0.142315	−	0.989821i	$-0.545455\pi$
$860$		0			0
$861$		0			0
$862$	1.44091	−	2.49574i	1.44091	−	2.49574i
$863$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$863$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$864$	−1.03809	−	0.989821i	−1.03809	−	0.989821i
$865$		0			0
$866$	0.947890	−	1.64179i	0.947890	−	1.64179i
$867$	0.580057	+	0.814576i	0.580057	+	0.814576i
$868$	1.24606	+	2.15824i	1.24606	+	2.15824i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.654861	−	0.755750i	$-0.727273\pi$
$877$	−0.981929	−	1.70075i	−0.981929	−	1.70075i	−0.327068	−	0.945001i	$-0.606061\pi$
$878$		0			0
$879$	0.195876	−	0.428908i	0.195876	−	0.428908i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.696647	+	2.01283i	−0.696647	+	2.01283i
$883$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$883$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	−1.39734	+	2.42027i	−1.39734	+	2.42027i
$890$		0			0
$891$	−1.65033	+	0.660694i	−1.65033	+	0.660694i
$892$	1.27051			1.27051
$893$	0.642315	−	1.11252i	0.642315	−	1.11252i
$894$		0			0
$895$		0			0
$896$	−0.215572	−	0.373381i	−0.215572	−	0.373381i
$897$		0			0
$898$	0.947890	−	1.64179i	0.947890	−	1.64179i
$899$	2.68757			2.68757
$900$	−0.358193	+	1.03493i	−0.358193	+	1.03493i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.415415	−	0.909632i	$-0.363636\pi$
$907$	0.995472	−	1.72421i	0.995472	−	1.72421i	0.580057	−	0.814576i	$-0.303030\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.654861	−	1.13425i	0.654861	−	1.13425i	−0.327068	−	0.945001i	$-0.606061\pi$
$911$	0.654861	−	1.13425i	0.654861	−	1.13425i	0.981929	−	0.189251i	$-0.0606061\pi$
$912$	1.75122	−	0.167221i	1.75122	−	0.167221i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.01671	+	1.76100i	−1.01671	+	1.76100i
$917$		0			0
$918$		0			0
$919$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$919$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$920$		0			0
$921$		0			0
$922$	0.947890	+	1.64179i	0.947890	+	1.64179i
$923$		0			0
$924$	−3.04678	+	0.290932i	−3.04678	+	0.290932i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−2.66323			−2.66323
$929$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	−0.959493	−	0.281733i	$-0.909091\pi$
$929$	−0.235759	+	0.408346i	−0.235759	+	0.408346i	0.723734	+	0.690079i	$0.242424\pi$
$930$		0			0
$931$	−1.44493	−	2.50268i	−1.44493	−	2.50268i
$932$	−0.258195	−	0.447206i	−0.258195	−	0.447206i
$933$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$934$	−1.34378	+	2.32750i	−1.34378	+	2.32750i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$	−1.61121	−	2.26262i	−1.61121	−	2.26262i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	0.415415	+	0.909632i	$0.363636\pi$
$947$	−0.580057	+	1.00469i	−0.580057	+	1.00469i	−0.995472	+	0.0950560i	$0.969697\pi$
$948$		0			0
$949$		0			0
$950$	−1.42131	−	2.46178i	−1.42131	−	2.46178i
$951$	1.98193	−	0.189251i	1.98193	−	0.189251i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.635257	+	1.10030i	−0.635257	+	1.10030i
$957$	−1.37115	+	3.00239i	−1.37115	+	3.00239i
$958$		0			0
$959$	0.0748038	+	0.129564i	0.0748038	+	0.129564i
$960$		0			0
$961$	−0.547582	+	0.948440i	−0.547582	+	0.948440i
$962$		0			0
$963$	0.514186	−	1.48564i	0.514186	−	1.48564i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	0.723734	−	0.690079i	$-0.242424\pi$
$967$	−0.235759	−	0.408346i	−0.235759	−	0.408346i	−0.959493	+	0.281733i	$0.909091\pi$
$968$	−0.148774	−	0.257684i	−0.148774	−	0.257684i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−0.973420	+	0.501833i	−0.973420	+	0.501833i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.372120	+	0.644532i	0.372120	+	0.644532i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	1.03115	−	1.78600i	1.03115	−	1.78600i
$980$		0			0
$981$		0			0
$982$	2.89494			2.89494
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	0.427201	−	0.935439i	0.427201	−	0.935439i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	1.03809	−	1.79803i	1.03809	−	1.79803i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.142315	+	0.989821i	$0.545455\pi$
$997$	−0.928368	+	1.60798i	−0.928368	+	1.60798i	−0.786053	+	0.618159i	$0.787879\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.1.f.b.1168.3	yes	20
9.4	even	3	inner	1503.1.f.b.166.3	✓	20
167.166	odd	2	CM	1503.1.f.b.1168.3	yes	20
1503.166	odd	6	inner	1503.1.f.b.166.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.1.f.b.166.3	✓	20	9.4	even	3	inner
1503.1.f.b.166.3	✓	20	1503.166	odd	6	inner
1503.1.f.b.1168.3	yes	20	1.1	even	1	trivial
1503.1.f.b.1168.3	yes	20	167.166	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1503.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.723734 + 1.25354i) q^{2} +(0.580057 + 0.814576i) q^{3} +(-0.547582 - 0.948440i) q^{4} +(-1.44091 + 0.137591i) q^{6} +(0.786053 - 1.36148i) q^{7} +0.137747 q^{8} +(-0.327068 + 0.945001i) q^{9} +O(q^{10})\)	\(q+(-0.723734 + 1.25354i) q^{2} +(0.580057 + 0.814576i) q^{3} +(-0.547582 - 0.948440i) q^{4} +(-1.44091 + 0.137591i) q^{6} +(0.786053 - 1.36148i) q^{7} +0.137747 q^{8} +(-0.327068 + 0.945001i) q^{9} +(0.888835 - 1.53951i) q^{11} +(0.454947 - 0.996196i) q^{12} +(1.13779 + 1.97070i) q^{14} +(0.447890 - 0.775768i) q^{16} +(-0.947890 - 1.09392i) q^{18} +1.96386 q^{19} +(1.56499 - 0.149438i) q^{21} +(1.28656 + 2.22839i) q^{22} +(0.0799009 + 0.112205i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.959493 + 0.281733i) q^{27} -1.72171 q^{28} +(-0.928368 + 1.60798i) q^{29} +(-0.723734 - 1.25354i) q^{31} +(0.717180 + 1.24219i) q^{32} +(1.76962 - 0.168978i) q^{33} +(1.07537 - 0.207261i) q^{36} +(-1.42131 + 2.46178i) q^{38} +(-0.945307 + 2.06993i) q^{42} -1.94684 q^{44} +(0.327068 - 0.566498i) q^{47} +(0.891724 - 0.0851493i) q^{48} +(-0.735759 - 1.27437i) q^{49} +(-0.723734 - 1.25354i) q^{50} +(0.341254 - 1.40667i) q^{54} +(0.108276 - 0.187540i) q^{56} +(1.13915 + 1.59971i) q^{57} +(-1.34378 - 2.32750i) q^{58} +(-0.415415 + 0.719520i) q^{61} +2.09516 q^{62} +(1.02951 + 1.18812i) q^{63} -1.18041 q^{64} +(-1.06891 + 2.34059i) q^{66} +(-0.0450525 + 0.130171i) q^{72} +(-0.995472 + 0.0950560i) q^{75} +(-1.07537 - 1.86260i) q^{76} +(-1.39734 - 2.42027i) q^{77} +(-0.786053 - 0.618159i) q^{81} +(-0.998692 - 1.40247i) q^{84} +(-1.84833 + 0.176494i) q^{87} +(0.122434 - 0.212062i) q^{88} +1.16011 q^{89} +(0.601300 - 1.31666i) q^{93} +(0.473420 + 0.819988i) q^{94} +(-0.595855 + 1.30474i) q^{96} +(-0.841254 + 1.45709i) q^{97} +2.12998 q^{98} +(1.16413 + 1.34347i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9	\( 20 q - q^{2} + q^{3} - 11 q^{4} - q^{6} - q^{7} - 2 q^{8} + q^{9} - q^{11} + q^{14} - 12 q^{16} + 2 q^{18} + 2 q^{19} - q^{21} + q^{22} + q^{24} - 10 q^{25} - 2 q^{27} - q^{29} - q^{31} - q^{33} + q^{38} - 13 q^{42} - 22 q^{44} - q^{47} + 21 q^{48} - 11 q^{49} - q^{50} - 12 q^{54} - q^{56} - q^{57} + q^{58} + 2 q^{61} + 42 q^{62} + 2 q^{63} + 20 q^{64} - 2 q^{66} - 10 q^{72} + q^{75} + q^{77} + q^{81} + 22 q^{84} - q^{87} - q^{88} + 2 q^{89} + 2 q^{93} + q^{94} + 2 q^{97} - 22 q^{98} + 2 q^{99}+O(q^{100}) \) 20 * q - q^2 + q^3 - 11 * q^4 - q^6 - q^7 - 2 * q^8 + q^9 - q^11 + q^14 - 12 * q^16 + 2 * q^18 + 2 * q^19 - q^21 + q^22 + q^24 - 10 * q^25 - 2 * q^27 - q^29 - q^31 - q^33 + q^38 - 13 * q^42 - 22 * q^44 - q^47 + 21 * q^48 - 11 * q^49 - q^50 - 12 * q^54 - q^56 - q^57 + q^58 + 2 * q^61 + 42 * q^62 + 2 * q^63 + 20 * q^64 - 2 * q^66 - 10 * q^72 + q^75 + q^77 + q^81 + 22 * q^84 - q^87 - q^88 + 2 * q^89 + 2 * q^93 + q^94 + 2 * q^97 - 22 * q^98 + 2 * q^99

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